---
title: "Background Mortality Adjustments"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Background Mortality Adjustments}
  %\VignetteEngine{knitr::rmarkdown}
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---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
```

## Why Background Mortality Matters

Health economic models track cohorts over long time horizons (often lifetime). Patients can die from the disease being modelled, but they also face **background (all-cause) mortality** from other causes. Census life tables provide general population mortality rates, but these must be adjusted for your specific patient population.

ParCC provides four methods for handling background mortality.

## Method 1: SMR Adjustment

### The Scenario  --  Heart Failure Cohort

You are modelling a cohort of 65-year-old heart failure patients. The general population mortality rate for age 65 (from the Indian Census life table) is **0.013 per year**. Published literature reports that heart failure patients have a **Standardized Mortality Ratio (SMR) of 2.5** compared to the general population (Meta-analysis by Jhund et al., Eur Heart J 2009).

### The Formula

$$r_{disease} = r_{population} \times SMR$$

$$p_{disease} = 1 - e^{-r_{disease} \times t}$$

### Worked Example

```{r}
r_pop <- 0.013   # General population rate, age 65
smr <- 2.5        # From published meta-analysis

r_disease <- r_pop * smr
p_disease <- 1 - exp(-r_disease)

cat("Population rate:", r_pop, "\n")
cat("SMR:", smr, "\n")
cat("Disease-adjusted rate:", round(r_disease, 5), "\n")
cat("Annual mortality probability:", round(p_disease, 5), "\n")
```

## Method 2: DEALE (Declining Exponential Approximation of Life Expectancy)

### The Scenario  --  Rare Disease with Known Life Expectancy

For Duchenne Muscular Dystrophy, the **observed mean life expectancy is 25 years** (from a registry study). The background life expectancy for the same age-sex group is **65 years**.

### The Formula

$$r_{disease} = \frac{1}{LE_{observed}} - \frac{1}{LE_{background}}$$

### Worked Example

```{r}
le_observed <- 25    # Disease-specific life expectancy
le_background <- 65  # General population

r_excess <- (1 / le_observed) - (1 / le_background)
cat("Excess mortality rate:", round(r_excess, 5), "\n")
cat("This means patients face an additional", round(r_excess * 1000, 2),
    "deaths per 1,000 person-years beyond background\n")
```

## Method 3: Gompertz Parameterization

### When to Use

The Gompertz law describes how mortality increases exponentially with age, which closely matches human mortality patterns above age 30:

$$r(age) = \alpha \times e^{\beta \times age}$$

This is useful when you need age-specific mortality rates and only have a few data points from the life table.

### Worked Example

```{r}
# Fit Gompertz from two life table points
# Age 60: rate = 0.008
# Age 80: rate = 0.065

r1 <- 0.008; age1 <- 60
r2 <- 0.065; age2 <- 80

beta <- log(r2 / r1) / (age2 - age1)
alpha <- r1 / exp(beta * age1)

cat("Gompertz alpha:", format(alpha, scientific = TRUE), "\n")
cat("Gompertz beta:", round(beta, 5), "\n\n")

# Generate age-specific rates
ages <- seq(50, 90, by = 5)
rates <- alpha * exp(beta * ages)
probs <- 1 - exp(-rates)

data.frame(
  Age = ages,
  Rate = round(rates, 5),
  Annual_Prob = round(probs, 5)
)
```

## Method 4: Linear Interpolation

For ages between available life table entries, linear interpolation provides a simple estimate:

$$r(age) = r_1 + (age - age_1) \times \frac{r_2 - r_1}{age_2 - age_1}$$

## References

- Beck JR, et al. A convenient approximation of life expectancy (the "DEALE"). *Am J Med*. 1982;73(6):883-888.
- Gompertz B. On the nature of the function expressive of the law of human mortality. *Phil Trans Royal Soc*. 1825;115:513-583.
- Jhund PS, et al. Long-term trends in first hospitalization for heart failure and subsequent survival. *Eur Heart J*. 2009;30(4):1807-1812.